THEORETICAL FOUNDATIONS OF FINANCIAL OPTIMIZATION: A RIGOROUS EXPLORATION OF THE HAMILTON-JACOBI-BELLMAN EQUATION
Abdulgaffar Muhammad , Anthony Unyime Abasido, Ibitomi TaiwoJohn Nma Aliu, Adedokun Lateef Adetunji, Maryam Isyaku and Anthony Kolade Adesugba
Volume 10 Issue 1
Abstract
The study provides a rigorous theoretical exploration of the Hamilton-Jacobi-Bellman (HJB) equation and its foundations in financial optimization. The HJB equation serves as a cornerstone in optimizing sequential decisionmaking under uncertainty. The study discusses the adaptation of the classical Hamilton-Jacobi equation from mechanics to finance, establishing the theoretical framework for applying dynamic programming in financial contexts. A comprehensive perspective on financial optimization as a mathematical discipline is presented, emphasizing the role of uncertainty and system dynamics. The integration of stochastic control theory is delineated, underscoring the elegance of adapting control concepts. The theoretical derivation and practical implications of the HJB equation are explicated in detail. The nonlinearities and challenges inherent in solving the HJB equation are also discussed. Keywords: Hamilton-Jacobi-Bellman equation, dynamic programming, stochastic control, financial optimization, nonlinear partial differential equations, viscosity solutions
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